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微積分數位化教材
C. L. Lang, Department of Applied Mathematics, I-Shou University
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指數定義
指數律
指數函數
C. L. Lang, Department of Applied Mathematics, I-Shou University
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
數的介紹中，數系的演進是
自然數 → 整數 → 有理數 → 實數
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對於指數的推演也是循類似的路徑 ：
自然指數 → 整數指數 → 有理指數
→ 實數指數。
C. L. Lang, Department of Applied Mathematics, I-Shou University
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
自然數指數定義
令𝑎≠0𝑎∈𝑅 𝑛∈𝑁
將 a 自乘 n 次記做 𝑎𝑛 。

例
𝑎2 = 𝑎 ∙ 𝑎
𝑎3 = 𝑎 ∙ 𝑎 ∙ 𝑎
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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
1.
2.
3.
指數律(Law of exponent)
令 𝑎，𝑏 ≠ 0 𝑎，𝑏 ∈ 𝑅 𝑚，𝑛 ∈ 𝑁 ，則
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
𝑎𝑚 𝑛 = 𝑎𝑚𝑛
𝑎𝑛 ∙ 𝑏𝑛 = (𝑎𝑏)𝑛
C. L. Lang, Department of Applied Mathematics, I-Shou University
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例
1.
2.
3.
32 ∙ 35 = 32+5 = 37
32 5 = 32∙5 = 310
32 ∙ 52 = 3 ∙ 5 2 = 152
C. L. Lang, Department of Applied Mathematics, I-Shou University
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1.
2.
3.
𝑎0 定義
若 𝑎0 是一個實數
符合指數律 𝑎𝑛 ∙ 𝑎0 = 𝑎𝑛+0
𝑎𝑛 = 𝑎𝑛+0 = 𝑎𝑛 ∙ 𝑎0
𝑎𝑛 = 𝑎𝑛 ∙ 1
所以
𝑎𝑛 ∙ 𝑎0 = 𝑎𝑛 ∙ 1
因此
𝑎0 = 1
C. L. Lang, Department of Applied Mathematics, I-Shou University
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定義
令 𝑎 ≠ 0 𝑎 ∈ 𝑅 ，則 𝑎0 = 1。
因為實數具有封閉性， 𝑎𝑛 𝑥 = 1 必有解，解為
1
𝑥= 𝑛
𝑎
C. L. Lang, Department of Applied Mathematics, I-Shou University
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整數指數定義
令 𝑎 ≠ 0 𝑎 ∈ 𝑅 𝑛 ∈ 𝑁，則
1
−𝑛
𝑎 = 𝑛
𝑎
所以對所有的整數 𝑛 ∈ 𝑍 ，𝑎𝑛 有定義如上。
C. L. Lang, Department of Applied Mathematics, I-Shou University
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指數律(Law of exponent)
令 𝑎，𝑏 ≠ 0 𝑎，𝑏 ∈ 𝑅 𝑚，𝑛 ∈ 𝑍 ，則
1. 𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛
2.
𝑎𝑚 𝑛 = 𝑎𝑚𝑛
3. 𝑎 𝑛 ∙ 𝑏 𝑛 = (𝑎𝑏)𝑛
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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例
1.
32 ∙ 3−5 = 32+(−5) = 3−3
2.
32 −5 = 32∙(−5) = 3−10
3−2 ∙ 5−2 = 3 ∙ 5 −2 = 15−2
3.
C. L. Lang, Department of Applied Mathematics, I-Shou University
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1.
2.
當 𝑎 > 0 𝑎 ∈ 𝑅 ，則 𝑥 𝑛 = 𝑎 必有解。
有理指數定義
令𝑎>0 𝑎∈𝑅
1
𝑛
定義 𝑎 為 𝑥 𝑛 = 𝑎 的正實數根。
1
𝑎𝑛
令𝑟∈
=
𝑛
𝑎
𝑚
𝑄，𝑟 = ，則
𝑛
𝑚
1 𝑚
𝑎𝑟 = 𝑎 𝑛 = 𝑎𝑛
=
𝑛
𝑎
C. L. Lang, Department of Applied Mathematics, I-Shou University
𝑚
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指數律(Law of exponent)
令 𝑎，𝑏 ∈ 𝑅 𝑎，𝑏 > 0 𝑟，𝑠 ∈ 𝑄 ，則
1. 𝑎 𝑟 ∙ 𝑎 𝑠 = 𝑎 𝑟+𝑠
2.
𝑎𝑟 𝑠 = 𝑎𝑟𝑠
3. 𝑎 𝑟 ∙ 𝑏 𝑟 = (𝑎𝑏)𝑟
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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例
2 ∙ 2=2 ∙2 =2
1.
2.
3.
1
3
1
2
3
3
3
2 =
3
2
1
3
1
3
= 2
1
3
1 1
+
2 3
1
1 3
2
5
6
=2 =
=2
2∙ 5=2 ∙5 = 2∙5
1
3
11
∙
23
6
25
=
6
6
2
1
6
=2 =
1
3
= 10 =
C. L. Lang, Department of Applied Mathematics, I-Shou University
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32
10
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對於實數指數我們利用逼近的方法定義
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實數指數定義
令 𝑎 > 0 𝑎，𝑥 ∈ 𝑅
假設 𝑟𝑛 是一連串逼近到 x 的有理數，則定義

𝑎 𝑥 就是 𝑎𝑟𝑛 的逼近值。
C. L. Lang, Department of Applied Mathematics, I-Shou University
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指數律(Law of exponent)
令 𝑎，𝑏 ∈ 𝑅 𝑎，𝑏 > 0 𝑥，𝑦 ∈ 𝑅 ，則
1. 𝑎 𝑥 ∙ 𝑎 𝑦 = 𝑎 𝑥+𝑦
2.
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦
3. 𝑎 𝑥 ∙ 𝑏 𝑥 = (𝑎𝑏) 𝑥
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由於 𝑎
−𝑦
=
1
𝑎𝑦
，所以
𝑎𝑥
1
𝑥
𝑥
−𝑦
𝑥−𝑦
=
𝑎
∙
=
𝑎
∙
𝑎
=
𝑎
𝑎𝑦
𝑎𝑦
C. L. Lang, Department of Applied Mathematics, I-Shou University
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1.
例
3
16 ∙
23
3
2.
16
8
1
1 2
2
=
4
23
3
24
8=
3
24
4
3
23
∙
4 3
+
3 4
3
4
=2 ∙2 =2
=2
4 3
+
3 4
=2
=
4 13
2
∙
25
12
=2
7
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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𝑎 𝑥 就是 𝑎𝑟𝑛 的逼近值。
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𝑎𝑟𝑛 > 0
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令 𝑎 > 0 ，對所有的 𝑥 ∈ 𝑅 ，則 𝑎 𝑥 > 0
C. L. Lang, Department of Applied Mathematics, I-Shou University
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解 22𝑥 − 2𝑥+1 = 8
由指數律
2𝑥 2 − 2 ∙ 2𝑥 − 8 = 0
所以
(2𝑥 − 4)(2𝑥 + 2) = 0
由於 2𝑥 + 2 > 2，所以
2𝑥 − 4 = 0
𝑥=2
C. L. Lang, Department of Applied Mathematics, I-Shou University
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解不等式 22𝑥 − 2𝑥+1 ≥ 8
由指數律
2𝑥 2 − 2 ∙ 2𝑥 − 8 ≥ 0
所以
(2𝑥 − 4)(2𝑥 + 2) ≥ 0
由於 2𝑥 + 2 > 2，所以
2𝑥 − 4 ≥ 0
𝑥≥2
C. L. Lang, Department of Applied Mathematics, I-Shou University
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
令𝑎>0
函數
𝑓 𝑥 = 𝑎𝑥
是一個以 a 為底的指數函數
(Exponential function with base a)。
C. L. Lang, Department of Applied Mathematics, I-Shou University
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注意下列敘述
函數 𝑓 𝑥 = 𝑎 𝑥 的定義域是整個實數R，值域
是 𝑥 ∈ 𝑅| 𝑥 > 0 。
對所有 𝑎 > 0 ，𝑓 0 = 1
當 𝑎 > 1 時，𝑓 𝑥 = 𝑎 𝑥 為遞增函數(Increasing
function)，當 0 < 𝑎 < 1 時，𝑓 𝑥 = 𝑎 𝑥 為遞減
函數(Decreasing function)。
當 𝑎 ≠ 1 函數 𝑓 𝑥 = 𝑎 𝑥 是一對一函數
C. L. Lang, Department of Applied Mathematics, I-Shou University
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指數函數的圖形
當𝑎>1時
C. L. Lang, Department of Applied Mathematics, I-Shou University
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指數函數的圖形
當0<𝑎<1時
C. L. Lang, Department of Applied Mathematics, I-Shou University
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任何指數函數的圖形都有y截距 1。
C. L. Lang, Department of Applied Mathematics, I-Shou University
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I-Shou University Department of Applied
Mathematics, C. L. Lang
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